A long while back, the code was 01, then (in preparation for expanding the number space,) Oftel split the 01 code to be 071 and 081, later on, large cities had an extra '1' inserted, so Birmingham became 0121, and London numbers 0171 (inner) and 0181 (outer).
Later on still, there was the introduction of completely new ranges of numbers such as 023, 024, 028 (NI), 029 (Wales, currently only in use in Cardiff) and for London, 020
That's 020 not 0207, 0208 (and even London's Evening Standard paper had it wrong when proclaiming "0203 is a new code for London").
Incidentally there are also 020 0xxx xxxx numbers (used for call centres and normally unknown, but reached via 0800, 0845, 0870, 03xxx dialling codes for businesses).
*So the actual range of numbers for London is currently:
020 0xxx xxxx
020 3xxx xxxx
020 7xxx xxxx
020 8xxx xxxx*
Initially the 7xxx and 8xxx series were still the "inner" and "outer" London areas, but later on, once the "7" and "8" were in place, it meant that 020 7123 and 020 8123 could be used, and be in completely random areas of Greater London (and while BT exchanges covered only small areas, the numbers from cable firms like Virgin Media can be spread over several miles as there's not the same geographic constraints).
Why am I bothered, you may ask?
Well, the original reason for splitting the dialling "area" code from the "local" part of the number was because anyone in the area can dial without needing to add the "area" code. So in London, you can dial a local number just dialling the last 8 digits (unless it is 020 0xxx xxxx - these are defined as "National dialling" only).
I'm bothered because it is really easy to show the number the correct way, and a matter of laziness and error (brought about by poor example in the media) which has led to most Londoners being unable to show their number in the correct form. When there's some problem overseas, the error is further promoted because the Foreign Office number shown on news broadcasts (020 7xxx xxxx) is shown as 0207 xxx xxxx so no wonder few people seem capable of getting it correct. I'm not saying they are "thick", just misinformed!
Few other places have this problem, though for a while, when some other numbers (Bristol and Reading, I think) were changed to 011x xxx xxxx there were examples with spurious digits in the wrong place (making the number, if dialled, either be unavailable, or a wrong number). That was down to residents just not following the examples (because Leicester, Sheffield, Nottingham and Leeds were not affected the same way).
I understand that when the '3' series runs out, there will be '6'xxx numbers, all still using the 020 area code. Only people outside need to dial 020, or, of course, someone using a mobile phone. I believe the media are responsible for confusing the public, mostly by showing numbers as 0207 or 0208 when the "gap" should have been present. TV news and adverts have continued to show the wrong form of number (ie they show 0207 xxx etc, not 020 7xxx) and what just "did it" for me was seeing a business (which is associated with Mumsnet users trialling its service) showing 0203 xxx xxxx as their help line number.
I know many people use mobile phones, so there's no "space" shown if you see a number displayed, but given there is a standard laid down, it makes most sense if everyone follows it.
Finally, some time back I did get DCMS (which is responsible for Media) to correct their Press Releases (to show 020 xxxx xxxx rather than the wrong form of number).
After all, if a Government department most closely connected with the perpetrators of misinformation are themselves "getting it wrong" then what hope that the advertising and media firms follow their bad habits, taking their (bad) example ?